(a) What is azimuthal quantum number? (b) Why did Sommerfeld introduce this quantum number?

The azimuthal quantum number, often denoted as 'l', is a quantum number that describes the angular momentum and shape of an atomic orbital. It is used to help define the wave function of an electron in an atom. The azimuthal quantum number can take on integer values, ranging from 0 up to n-1, where 'n' is the principal quantum number. Each value of 'l' corresponds to a specific orbital shape, such as s, p, d, and f orbitals for l = 0, 1, 2, and 3, respectively.

Arnold Sommerfeld introduced the azimuthal quantum number as an extension to the Bohr model of the atom, which was proposed by Niels Bohr in 1913. The Bohr model explained the quantization of energy levels by means of a principal quantum number 'n' but could not account for the fine structure observed in atomic emission and absorption spectra. Sommerfeld's modification to the Bohr model added elliptical orbits and an angular momentum description, requiring the introduction of the azimuthal quantum number 'l' to help explain the observed fine structure. The azimuthal quantum number incorporated the concept of multi-electron atoms and allowed for a more accurate representation of the subshells and orbital shapes in various elements. This paved the way for the development of modern quantum mechanics and atomic theory.